1, 1, 2, 4, 8, ?, ?, 56, 108, 208, 401

The two ones at the beginning are usually seen in Fibonacci-like sequences. 

Trying taking the sum of the prevoius three terms did not work, but taking the sum of four prevoius terms does generate the sequence:
1, 1, 2, 4, 8, 15, 29, 56, 108, 208, 401, 773, 1490, . . . .